PRECALC CHAPTER 1.6 – INVERSE FUNCTIONS – NOTES
Domain: set of X values
Range: set of Y values
Relation: A set of ordered pairs – no boundaries
Function: (Also called a mapping) A relation between two sets in which one element of
the second set is assigned to each element of the first set OR a set of ordered pairs in
which none of the first elements of the pairs appears twice.
To determine a function –
1. Ordered pairs - no repeating x ’s
2. Graphs or equations ( y = mx + b ) – Vertical line test
Inverse Function: range becomes the domain and the domain becomes the range.
Notation for inverse: y −1 or f −1 ( x)
(The domain of f is equal to the range of f −1 and the range of f is equal to the domain
of f −1 . The graph of f −1 is a reflection of the graph of f over the line y = x .)
One-to-One function: Each element in the second set corresponds to ONLY one element
in the first set
•
In order for a function to have an inverse “function”, it must be one-to-one;
the original must pass the horizontal line test.
For f (x) and g (x) to be inverses of each other, f ( g ( x)) = g ( f ( x)) = x .
•
Notations:
f (x) ‘f of x’
f −1 ( x) inverse
1
f −1 ( x) ≠
f ( x)
Finding Inverse Functions:
A. Swap ordered pairs
ex 1: Given function f, find the inverse. Is the inverse also a function?:
Function f is a one-to-one function since the x and y values are used only once. The
inverse is:
Since function f is a one-to-one function, the inverse is also a function.
ex 2: Determine the inverse of this function. Is the inverse also a function?
x
1
-2
-1
0
2
3
4
-3
f(x)
2
0
3
-1
1
-2
5
1
Swap the x and y variables to create the inverse. Since function f was not a one-to-one
function (the y value of 1 was used twice), the inverse will NOT be a function (because
the x value of 1 now gets mapped to two separate y values which is not possible for
functions).
x
2
0
3
-1
1
-2
5
1
-1
f (x) 1
-2
-1
0
2
3
4
-3
ex 3:
1. f ( x) = x 3
D
-2
-1
0
1
2
R
-8
-1
0
1
8
Function?
One-to-one?
2. f ( x) = x 2
D
-2
-1
0
1
2
R
4
1
0
Function?
One-to-one?
3. f ( x) = x
D
R
0
0
-1
1
1
4
-2
9
2
-3
3
Function?
One-to-one?
B. Solve algebraically
• Set the function = y
• Swap the x and y variables
• Solve for y
ex 3: Find the inverse of the function f ( x) = x − 4
Set = y
y = x−4
Swap variables
x = y−4
Solve for y
x+4= y
f −1 ( x) = x + 4
x +1
ex 4: f ( x) =
x
EX5: Verify f ( x) = 2 x 3 − 1 and g ( x) = 3
x +1
are inverses of each other.
2
x +1
in calculator. Graph line y = x .
2
Look at a few points on f (x) . Compare these points to some points on g (x) .
What do you notice?
Summary:
• Only one-to-one functions have unique inverses
• Domain of function = range of inverse
Range of function = domain of inverse
• An inverse is a reflection over the line y = x
• Inverse of a function may not always be a function – the original must be
one-to-one to guarantee that its inverse will always be a function
EX6: Graph f ( x) = 2 x 3 − 1 and g ( x) = 3